The second fundamental theorem of invariant theory for the orthogonal group

Abstract

Let $V=\mathbb{C}^n$ be endowed with an orthogonal form and $G=\mathrm{O}(V)$ be the corresponding orthogonal group. Brauer showed in 1937 that there is a surjective homomorphism $\nu:B_r(n)\to\mathrm{End}_G(V^{\otimes r})$, where $B_r(n)$ is the $r$-string Brauer algebra with parameter $n$. However the kernel of $\nu$ has remained elusive. In this paper we show that, in analogy with the case of $\mathrm{GL}(V)$, for $r\geq n+1$, $\nu$ has a kernel which is generated by a single idempotent element $E$, and we give a simple explicit formula for $E$. Using the theory of cellular algebras, we show how $E$ may be used to determine the multiplicities of the irreducible representations of $\mathrm{O}(V)$ in $V^{\otimes r}$. We also show how our results extend to the case where $\mathbb{C}$ is replaced by an appropriate field of positive characteristic, and we comment on quantum analogues of our results.